208 IX ANALYTIC FUNCTIONS

containing [0, 1]. Let A be the closed subset of the interval [0, 1] consisting
of the t such that h(s) = 0 for 0 ^ s < t; by assumption there is an open
neighborhood of 0 in [0, 1] which is contained in A, hence the l.u.b. p of A
is certainly >0; we will prove that p = 1, which will establish (9.4.1). Note
first that h(f) = 0 for 0 < t < p, hence by continuity h(p) = 0; as h is analytic
at the point p, there is a power series in t - p, which converges for \t - p\ < a,
with a > 0 and whose sum is equal to h(t) for \t — p\ <a. But for 0 ^ / < p,
h(t) = 0 by assumption; from the principle of isolated zeros (9.1.5) it follows
that h(t) = Q for \t — p\ <a, which would contradict the definition of p
if we had p < 1.

(9.4.2) ("Principle of analytic continuation") Let A c= Kp be an open
connected set, f and g two analytic functions in A with values in E. If there
is a nonempty open subset VofA such thatf(x) = g(x) in U, thenf(x) = g(x)
for every x e A.

Let B be the interior of the set of points x e A such that f(x) = g(x).
It is clear that B is open and nonempty by assumption; we prove that B
is also closed in A, hence equal to A since A is connected (see (3.19)). Let
a G A be a cluster point of B; as/, g are analytic, there is an open polydisk
P of center a, contained in A, such that in P, f(x) and g(x) are equal to the
sums of two power series in the xt — at, absolutely summable in P. But by
definition, P n B contains an open polydisk U in which f(x) = g(x). By
(9.4.1) applied to P = Q, we conclude that/(;c) = g(x) in P, in other words
P c B, and in particular a e B. Q.E.D.

For;? = 1, we can improve (9.4.2) as follows:

(9.4.3) Let A c K be an open connected subset of K, / and g two analytic
functions in A with values in E. Suppose there is a compact subset E of A
such that the set M of points xeHfor which f(x) = g(x) be infinite. Then
f(x) = g(x)for every xeA.

Let (zn) be an infinite sequence of distinct points of M; as H is compact,
there is a cluster value b e H for the sequence (zn), hence any ball P of center
b, contained in A, contains an infinity of points of M. But we can suppose
/ and g are equal to convergent power series in z — b in a ball P c A of
center b; the principle of isolated zeros (9.1.6) then shows that/(jc) = #(*)
in P, and we can then apply (9.4.2).

For K = C we can also improve (9.4.2) in the following way:r series